Vector Equations of Lines, Planes, and Conics - Vectors (Undergraduate Foundation)

This course explores a crucial application of vector analysis: describing and solving problems in geometry. We begin by mastering the techniques for solving abstract vector equations before applying these skills to define geometric shapes. You will learn to construct the vector equations for lines, planes, and the conic sections (circles, ellipses, parabolas, and hyperbolas) in both two and three dimensions. The ability to describe complex geometries with concise equations is a cornerstone of modern science and engineering. This course bridges the gap between abstract vector theory and its practical application in modeling the real world. By working through a vast library of examples, you will learn to analyze the relationships between shapes—calculating intersections, angles, and distances—with precision and confidence. This advanced course is designed for students ready to apply their knowledge of vector products to analytical geometry. A thorough understanding of vector algebra, dot products, and cross products is essential. This programme is the critical bridge to higher-level topics, making it ideal for university students preparing for courses in vector calculus or mechanics.

2

Enrolment valid for 12 months

Course Chapters

1. Introduction
1

Welcome and course outline.

Chapter lessons

1-1. Welcome
10:41

Welcome to the course and outline of course.

2. Solving Vector Equations
1
8

Solutions of vector equations with unknown vectors; solutions of vector equations with unknown scalars.

Chapter lessons

2-1. Techniques
44:43

An overview of the solution techniques for vector equations with unknown vectors or unknown scalars.

3. Straight Lines
6
9

Direction vector and vector equation of a straight line,; angle between two straight lines; intersecting lines, parallel lines and skew lines; shortest distance between two skew lines.

Chapter lessons

3-1. Introduction
14:07

An introduction to the vector equations of geometries.

3-2. Direction and one point
21:53

Vector equation of a straight line through a given point in a given direction.

3-3. Two points
21:23

Vector equation of a straight line through two given points.

3-4. Angle between two lines

How to find the angle between two lines from their vector equations.

3-5. Intersection

Intersection of two lines; intersection of a line with a plane.

3-6. Skew lines

Meaning of skew lines and the perpendicular distance between two skew lines.

4. Planes
7
8

Vector equations of a plane; normal vector of a plane; parallel planes and the distance between them; angle between two planes; angle between a line and a plane.

Chapter lessons

4-1. A point and two lines

Vector equation of a plane through a given point and parallel to two given lines.

4-2. Three non-collinear points

Vector equation of a plane containing three given non-collinear points.

4-3. A point and a normal vector

Vector equation of a plane containing a given point and normal to a given vector (or line).

4-4. Parallel planes

Parallel planes, their equations and minimum (perpendicular) distance apart.

4-5. Angle between two planes

How to find the angle between two intersecting planes.

4-6. Intersection with a line

Re-examining the point of intersection of a line and a plane; angle between a line and a plane.

4-7. Intersection of two planes

How to find the equation of the line of intersection of two planes.

5. Circles
3
3

Vector equations of a circle in x-y plane, a circle in space; non-parametric vector and standard forms; equation of the plane containing a circle.

Chapter lessons

5-1. X-y plane

Vector equation of a circle in x-y plane.

5-2. Three dimensions

Vector equation of a circle in a three-dimensional space.

5-3. Non-parametric equation

General non-parametric of equation of a circle in a plane; non-parametric of equation of a circle in the x-y plane.

6. Ellipses
3
2

Parametric and non-parametric vector equations of an ellipse.

Chapter lessons

6-1. X-y plane

Parametric equation of an ellipse in the x-y plane.

6-2. Three dimensions

Parametric equation of an ellipse in three dimensions.

6-3. Non-parametric equations

Non-parametric equations of an ellipse in two and three dimensions.

7. Parabolas
3
2

Parametric and non-parametric vector equations of a parabola.

Chapter lessons

7-1. X-y plane

Parametric vector equations of a parabola in the x-y plane.

7-2. Three dimensions

Parametric vector equations of a parabola in 3 dimensions.

7-3. Non-parametric equations

Non-parametric equations of a parabola in two and three dimensions.

8. Hyperbolas
3
2

Parametric and non-parametric vector equations of a hyperbola.

Chapter lessons

8-1. X-y plane

Parametric vector equations of a hyperbola in the x-y plane.

8-2. Three dimensions

Parametric equation of a hyperbola in three dimensions.

8-3. Non-parametric equations

Non-parametric equations of a hyperbola in two and three dimensions.